Maybe you can go somewhere close by? Like hang out at a mall or hotel even? That way you can labor close by, and can be there in a minute or two if need be. We're a tiny bit in the same situation, I didn't have a super fast labor but I'll be giving birth in the dead of winter in snow country with the hospital and hr and a half away. We plan on leaving as soon as labor hits or getting a hotel if a blizzard is expected. I'd rather be overly cautious than birth a baby in the car.

And in addition to the "oops" kit we can probably just head over to the hospital campus without going in. It's a teaching hospital university campus with a lot of campus as well as a huge walking/biking park/path and tons of wooded recreational area close by. We've talked about packing up and going over early and doing a walk or just hanging out nearby. There's also a big neighborhood with lots of restaurants and pretty walking scenery down the hill.